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Money and Price Posting Under Private Information!

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Money and Price Posting Under Private Information! Money and Price Posting under Private Information Mei Dongy Janet Hua Jiangz September 16, 2010 Abstract We study price posting with undirected search in a search theoretic model with divisible money and divisible goods. Ex ante homogeneous buyers experience match speci…c preference shocks in bilateral trades. The shocks follow a continuous distribution and the realization of the shocks is private information. We show that generically there exists a unique price posting monetary equilibrium. In equilibrium, each seller posts a continuous pricing schedule that exhibits quantity discounts. Buyers spend only when they have high preferences, and as preferences are higher, they spend more. As in‡ation decreases the value of waiting for future trades, higher in‡ation induces more buyers to spend money and those who spend before to spend their money faster. The model naturally captures the hot potato e¤ect of in‡ation along both the intensive margin and the extensive margin. Keywords: Price posting, Undirected search, Private information, Hot potato e¤ect, In‡a- tion, Quantity discounts JEL: D82, D83, E31 We thank Aleksander Berentsen, Jose Dorich, Huberto Ennis, Ben Fung, Peter Norman, Randy Wright, and seminar participants at the Bank of Canada, the 2009 Canadian Economic Association Meetings in Toronto, the 2009 Chicago Fed Summer Workshop on Money, Banking and Payments, the 2010 Midwest Macro Meetings, the 2010 Econometric Society World Congress in Shanghai, and the University of Manitoba Delta Marsh Conference on Monetary Economics for their comments and suggestions. The views expressed in this paper are those of the authors. No responsibility for them should be attributed to the Bank of Canada. ythe Bank of Canada; email: [email protected] zDepartment of Economics, University of Manitoba; email: [email protected] 1 1 Introduction This paper builds a search theoretic monetary model based on the framework developed by Lagos and Wright (2005) and Rocheteau and Wright (2005), where sellers post prices and buyers engage in undirected search in the sense that buyers only see the posted prices after they are matched with sellers. Price posting with undirected search (hereafter price posting) is an interesting pricing mechanism to study because it captures the characteristics of many daily exchanges.1 In many occasions, buyers randomly enter a store, read the prices and decide whether or not to make a purchase. In the search theoretic monetary literature, very few papers have studied price posting because it is hard to generate equilibria with valued …at money under this pricing mechanism. Generally, the existence of monetary equilibria hinges critically on the condition that the buyer extracts some surplus through trading with the seller. In a typical monetary model with price posting, sellers can extract all the trading surplus by proposing prices and quantities, and buyers decide whether to trade or not. In this case, sellers have all the bargaining power so that in equilibrium no buyers want to hold money when the nominal interest rate is positive. Monetary equilibrium thus unravels under price posting. To generate monetary equilibria under price posting, we introduce match speci…c preference shocks which a¤ect buyers’ marginal willingness to consume and private information about the realization of the shocks. When the preference shocks follow a continuous distribution, we show that there exists a unique monetary equilibrium. The model is then used to examine the seller’s pricing schedule, the buyer’s spending pattern and the e¤ects of in‡ation. We …nd that in equilibrium, sellers post a non-linear pricing schedule and the unit price exhibits quantity discounts: the unit price is lower for purchases of larger sizes. Quantity discounts are frequently observed in practice and the traditional explanation is that the unit cost of producing a large quantity is lower. In our model, quantity discounts exist even when the unit cost is constant. It is used by sellers to induce buyers with higher preferences to consume more. Buyers spend money only when they have high enough marginal utilities and spend more when their marginal utilities are higher. As for the e¤ects of in‡ation, we show that buyers spend their money faster when in‡ation is higher; this is the "hot potato" e¤ect of in‡ation. The intuition is that in‡ation reduces the bene…t of retaining money for future trading opportunities and induces buyers to spend their money faster. 1 We use price posting to refer to price posting with undirected search in the rest of the paper. When price posting is combined with directed search, it is generally labelled as competitive search in the literature. 2 More importantly, price posting equilibrium endogenously generates the hot potato e¤ect along both the intensive (those who spend choose to spend more) and the extensive margins (those who do not spend start spending) without resorting to free entry or search intensity. There have been a few papers that study price posting. As in our paper, the key to the existence of a monetary equilibrium under price posting is private information about match speci…c preference shocks. Jafarey and Masters (2003) and Curtis and Wright (2004) are built on the indivisible money model of Trejos and Wright (1995). In Curtis and Wright (2004), there are K 2 realizations of the preference shock and in equilibrium, sellers post at most two prices. In Jafarey and Masters (2003), the preference shock follows a uniform distribution. Interestingly, there is only a single price posted in equilibrium. In general, there exists the law of two prices in models with indivisible money as emphasized by Curtis and Wright (2004). More recently, Ennis (2008) extends price posting to a divisible money framework as in Lagos and Wright (2005). He speci…es a distribution of preference shocks that takes two values, and shows that sellers post a single price in equilibrium. The above three papers generate monetary equilibria under price posting, however, they are not suitable to analyze sellers’pricing behavior and buyers’spending pattern and how they are a¤ected by in‡ation. In equilibrium, at most two prices are posted, and buyers spend all their money during exchanges. In our model, money is divisible and preference shocks follow a continuous distribution. The model implies richer (and more realistic) pricing behavior and allows us to study the e¤ect of in‡ation on buyers’spending pattern in a natural environment. Several recent papers have tried to rationalize the hot potato e¤ect of in‡ation using microfounded monetary theory. Lagos and Rocheteau (2005) endogenize search intensity, and in‡ation induces buyers to search more intensively. The explanation, however, is not robust and only applies to competitive search at low in‡ation. In Ennis (2009), sellers have more opportunities to rebalance money holdings than buyers, in‡ation makes buyers search harder to …nd sellers to o¤-load their cash. Liu et al. (forthcoming) o¤er another explanation. They assume that buyers pay the entry fee to enter the market. In‡ation reduces buyers’trading surplus, so fewer buyers choose to enter, which increases the matching probability for those who do enter. Nosal (forthcoming) assumes that accepting a current trade reduces the probability of future trading. In‡ation reduces the value of future trading and buyers are more likely to accept current trades. The rest of the paper proceeds as follows. Section 2 describes the environment. In Section 3, we characterize the monetary equilibrium when the preference shock follows a uniform distribution. Section 4 examines the e¤ect of in‡ation and rationalizes the hot potato e¤ect of in‡ation. In Section 5, the model is extended to allow for a more general continuous distribution of the preference 3 shock. Finally, Section 6 concludes. The technical proofs of results in Section 5 are provided in the Appendix. 2 Environment The model is based on Lagos and Wright (2005). Time is discrete and runs from 0 to .A 1 decentralized market (DM) and a centralized market (CM) open sequentially in each period. The discount factor between two periods is 0 < < 1. There are two permanent types of agents: buyers and sellers distinguished by their roles in the DM. There is one nonstorable good in each submarket: a CM good and a DM good. The CM is a centrally located competitive spot market. In the CM, all agents can consume or produce the CM good x. The utility of consuming x units of the CM good is x. If x < 0, it means that an agent produces x units of the CM good and incurs disutility x. In the DM, agents are anonymous. Buyers and sellers are randomly matched so that one buyer meets one seller with probability 1. Buyers are those who want to consume but cannot produce. Sellers, on the other hand, can produce but do not want to consume. This generates a lack of double coincidence of wants problem. Together with anonymity, money becomes essential as the medium of exchange. For a seller, the disutility of producing q units of the DM good is c(q) with c(0) = 0, c0 > 0, and c00 0. By consuming q units of the DM good, a buyer’sutility is eu(q) where e 0 is a preference parameter that determines the buyer’s marginal utility of consumption. The utility function u(q) satis…es u(0) = 0, u0 > 0 > u00 and u0(0) = . 1 All buyers are ex ante identical before being matched with sellers. Ex post, however, they are subject to match speci…c preference shocks and become heterogeneous during their matches with sellers. The realization of e follows a uniform distribution on the interval [0; 1]. Buyers hold private information about the realization of e.
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